Daily Question 105

Daily Question 105 

$\textbf{Q105.}$ Let $f(x)=\begin{vmatrix} x & 1 & 1\\ \sin\pi x & 2x^2 & 1\\ x^3 & 3x^4 & 1\\ \end{vmatrix}$. If $f(x)$ be an odd function and its odd value is equal to $g(x)$ then $f(1)g(1)$ is

(A) $-1$

(B) $-5$

(C) $1$

(D) $-4$

$\textbf{Ans.} (D)$

$\textbf{Sol.}$ Given $f(x)$ is an odd function, hence $g(x)=f(-x)=-f(x)$

Now, $f(x)g(x)=-f(x) \cdot f(x)$

$f(x)g(x)=-\begin{vmatrix} x & 1 & 1\\ \sin\pi x & 2x^2 & 1\\ x^3 & 3x^4 & 1\\ \end{vmatrix}\begin{vmatrix} x & 1 & 1\\ \sin\pi x & 2x^2 & 1\\ x^3 & 3x^4 & 1\\ \end{vmatrix}$

$f(x)g(x)=-\begin{vmatrix} x^2 + \sin\pi x + x^3 & x + 2x^2 + 3x^4 & x + 2\\ x\sin\pi x + 2x^2\sin\pi x + x^3 & \sin\pi x + 7x^4 & \sin\pi x + 2x^2 +1\\ x^4 + 3x^4\sin\pi x + x^3 & x^3 + 6x^6 +3x^4 & x^3 + 3x^4 +1\\ \end{vmatrix}$

$f(1)g(1)=-\begin{vmatrix} 2 & 6 & 3\\ 1 & 7 & 3\\ 2 & 10 & 5\\ \end{vmatrix}$

$f(1)g(1)=-4$

Compiled Daily Questions 1-100